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Properties

Label 2-384-12.11-c1-0-14
Degree $2$
Conductor $384$
Sign $-0.356 + 0.934i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.618 − 1.61i)3-s − 3.23i·5-s + 1.23i·7-s + (−2.23 − 2.00i)9-s + 5.23·11-s − 4.47·13-s + (−5.23 − 2.00i)15-s − 2.47i·17-s + 0.763i·19-s + (2.00 + 0.763i)21-s − 2.47·23-s − 5.47·25-s + (−4.61 + 2.38i)27-s − 4.76i·29-s + 5.23i·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s − 1.44i·5-s + 0.467i·7-s + (−0.745 − 0.666i)9-s + 1.57·11-s − 1.24·13-s + (−1.35 − 0.516i)15-s − 0.599i·17-s + 0.175i·19-s + (0.436 + 0.166i)21-s − 0.515·23-s − 1.09·25-s + (−0.888 + 0.458i)27-s − 0.884i·29-s + 0.940i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.356 + 0.934i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821977 - 1.19386i\)
\(L(\frac12)\) \(\approx\) \(0.821977 - 1.19386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.618 + 1.61i)T \)
good5 \( 1 + 3.23iT - 5T^{2} \)
7 \( 1 - 1.23iT - 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 - 0.763iT - 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 4.76iT - 29T^{2} \)
31 \( 1 - 5.23iT - 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 3.23iT - 53T^{2} \)
59 \( 1 + 1.23T + 59T^{2} \)
61 \( 1 - 0.472T + 61T^{2} \)
67 \( 1 - 9.70iT - 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 0.291iT - 79T^{2} \)
83 \( 1 - 2.76T + 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 - 0.472T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.59955937811697387787218550642, −9.700004049350334869849213063187, −9.123085472392491002397589723884, −8.375311004136550745503923058818, −7.38768480962131930711332998032, −6.31772620663705454161691159052, −5.24017022313894892514145531576, −4.06233545444411783821465339472, −2.34162863280202814876175064112, −0.990795964148147302576711378088, 2.41150850783916056386223581536, 3.59307890147807467427570262616, 4.39857266292059540737170685877, 5.95528125231371632951653869061, 6.93886039116066102818692117218, 7.81355643235460799238542660951, 9.169719782415862563373545761261, 9.835940482815470902539602454077, 10.66935905546697630113358430958, 11.31235256190131899215838706657

Graph of the $Z$-function along the critical line